
\(L_{pq}=\mathbb {Q}(\sqrt{2}, \sqrt{qp}, \sqrt{-1}) \) .

\(L_{pq}^*=\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{-1})\)  the genus field of \(L_{pq}\) .

\(L_{p^{\prime }}=\mathbb {Q}(\sqrt{2}, \sqrt{p^{\prime }}, \sqrt{-1}) \) .

\(F_{pq}=\mathbb {Q}(\sqrt{2p}, \sqrt{2q}, \sqrt{-2})\)  or \(\mathbb {Q}(\sqrt{p}, \sqrt{2q}, \sqrt{-2})\) , according to whether \(p\)  and \(q\)  verify conditions (REF ) or (REF ) .

\(K_{pq}=\mathbb {Q}(\sqrt{p}, \sqrt{q}, \sqrt{-2})\)  or \(\mathbb {Q}(\sqrt{q}, \sqrt{2p}, \sqrt{-2})\) , according to whether \(p\)  and \(q\)  verify conditions (REF ) or (REF ).

\(k_{pq}=\mathbb {Q}(\sqrt{-2}, \sqrt{pq})\)  or \(\mathbb {Q}(\sqrt{-2}, \sqrt{2pq})\)  according to whether \(p\)  and \(q\)  verify conditions (REF ) or (REF ).

\(m\ge 2\)  the positive integer satisfying \(h_2(-2q)=2^m\)  (cf. [1]}).

